On a sufficient condition for function to be p -valent close-to-convex
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On a sufficient condition for function to be p-valent close-to-convex Mamoru Nunokawa1 · Janusz Sokół2
· Lucyna Trojnar-Spelina3
Received: 9 May 2018 / Accepted: 21 June 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract The property of close-to-convexity of analytic function, that generalizes that of starlikeness, was introduced by Kaplan in 1952. He also gave the geometric interpretation of this property and proved that all close-to-convex functions in the unit disc D are univalent. Another geometric interpretation was given later by Lewandowski. In this paper, we established some sufficient conditions for functions analytic in the unit disc D to be p-valently close-to-convex in D. Keywords Univalent functions · Starlike · Convex · Close-to-convex Mathematics Subject Classification Primary: 30C45 · Secondary: 30C80
1 Introduction A function f analytic in a domain D ∈ C is called p-valent in D, if for every complex number w, the equation f (z) = w has at most p roots in D, so that there exists a complex number w0 such that the equation f (z) = w0 has exactly p roots in D. We denote by H the class of functions f (z) which are holomorphic in the open unit disc
B
Janusz Sokół [email protected] Mamoru Nunokawa [email protected] Lucyna Trojnar-Spelina [email protected]
1
University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba 260-0808, Japan
2
Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland
3
Faculty of Mathematics and Applied Physics, Rzeszów University of Technology, Al. Powsta´nców Warszawy 12, 35-959 Rzeszów, Poland
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M. Nunokawa et al.
D = {z ∈ C : |z| < 1}. Denote by A p , p ∈ N = {1, 2, . . .}, the class of functions f (z) ∈ H given by ∞ f (z) = z p + an z n , (z ∈ D). n= p+1
Let A = A1 . Let S denote the class of all functions in A which are univalent. Also let S ∗p and C p be the subclasses of A p defined as follows: z f (z) > 0, z ∈ D , = f (z) ∈ A p : Re f (z) z f (z) C p = f (z) ∈ A p : Re 1 + > 0, z ∈ D . f (z)
S ∗p
The classes S ∗p and C p will be called the class of p-valently starlike functions and the class of p-valently convex functions, respectively. Note that S1∗ = S ∗ and C1 = C, where S ∗ and C are usual classes of starlike and convex functions, respectively.
2 Preliminaries In this paper, we need the following lemmas. Lemma 2.1 [10, Thm. 5] If f (z) ∈ A p , then for all z ∈ D, we have Re
z f ( p) (z) f ( p−1) (z)
>0
⇒
∀k ∈ {1, . . . , p} :
Re
z f (k) (z) f (k−1) (z)
> 0.
(2.1)
n Lemma 2.2 [11] Let q(z) = 1 + ∞ n=1 cn z , be analytic function in |z| < 1 with q(z) = 0. If there exists a point z 0 , |z 0 | < 1, such that |arg {q(z)} |
0, then we have 2ik arg {q(z 0 )} z 0 q (z 0 ) = , q(z 0 ) π for some k ≥ (a + a −1 )/2 ≥ 1, where {q(z 0 )}1/β = ±ia, and a > 0.
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On a sufficient condition for function to be p-valent close-to-convex
Lemma 2.3 [9,
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